Reverse mathematics, ordinal numbers, and the ACC

A Survey of the Reverse Mathematics of Ordinal Arithmetic

This article surveys theorems of reverse mathematics concerning the comparability, addition, multiplication and exponentiation of countable well orderings. In [13], Simpson points out that ATR0 is “strong enough to accommodate a good theory of countable ordinal numbers, encoded by countable well orderings.” This paper provides a substantial body of empirical evidence supporting Simpson’s claim....

Reverse mathematics and the ACC

On orders induced by implications

In this paper, the orders induced by the residual implications obtained from uninorms are investigated. A necessary and sufficient condition is presented so that the ordinal sum of fuzzy implications satisfies the law of importation with a t-norm $T$. Some relationships between the orders induced by an ordinal sum implication and its summands are determined. The algebraic structures obtained fr...

Epsilon Numbers and Cantor Normal Form

An epsilon number is a transfinite number which is a fixed point of an exponential map: ω ε = ε. The formalization of the concept is done with use of the tetration of ordinals (Knuth's arrow notation, ↑↑). Namely, the ordinal indexing of epsilon numbers is defined as follows: ε0 = ω ↑↑ ω, εα+1 = εα ↑↑ ω, and for limit ordinal λ: ε λ = lim α<λ εα = α<λ εα. Tetration stabilizes at ω: α ↑↑ β = α ↑...

Inner models and large cardinals

§0. The ordinal numbers were Georg Cantor’s deepest contribution to mathematics. After the natural numbers 0, 1, . . . , n, . . . comes the first infinite ordinal number ù, followed by ù + 1, ù + 2, . . . , ù + ù, . . . and so forth. ù is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identifi...